Introduction to the integral of 1/sqrt(x)

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function. 

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to an algebraic function 1/√x. You will also understand how to compute 1/√x integral by using different integration techniques.

What is the integral of 1/sqrt x?

The integral of 1/√x is an antiderivative of 1/sqrt x function which is equal to 2√x. It is also known as the reverse derivative of the function 1/√x which is an algebraic function.

We can integrate 1/sqrt x by using the power rule of the integral. This rule is written as;

$\int x^n dx=\frac{x^{n+1}}{n+1}+c$

This formula says that the integral of any algebraic function with some exponent, an be calculated by adding 1 in its exponent and dividing by the new exponent i.e n+1.

Integration of 1/√x formula

The formula of integral of square root x squared contains the integral sign, coefficient of integration and the function as 1/√x. It is denoted by ∫1/√x dx. In mathematical form, the integral of 1/√x is:

$\int \frac{1}{\sqrt{x}}dx=2\sqrt{x}+c$

 Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Similarly, the integral of sqrtx is 2√x +c.

How to calculate the integral 1/root x?

The integration of square root formula is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of a 1/sqrtx by using:

1. U-substitution method
2. Integration by parts
3. Definite integral

Integral of 1/sqrt(x) by using u-substitution method

The u-substitution is a method of integration in calculus. It is used to calculate the integral of a function which is complex to be calculated by usual integration. Let’s discuss calculating the antiderivative of 1/sqrt x by using u-substitution.

Proof of Integral of 1/√x by using substitution method

To prove the integral of 1/√x by using the substitution method, suppose that:

$I=\int \frac{1}{\sqrt{x}}dx$

Suppose that,

$u=\sqrt{x}$

and

$du=\frac{1}{2\sqrt{x}}dx$

Since u=√x,

So,

$dx=2udu$

Now using this value in the above integral, 

$I=2\int \frac{u}{u}du$

We can write it as;

$I=2\int du$

Integrating with respect to u,

$I=2u+c$

Now substituting the value of u,

$I=2\sqrt{x}+c$

Hence we have verified the x square root integral by using the u-substitution method calculator.

Integral of 1/sqrt x by using power rule

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The power rule of integration is a method of solving the integral of algebric functions with some exponent. Let’s discuss calculating the integral of 1/sqrt(x) by using power rule.

Proof of integral of 1/sqrtx by using power rule

To integrate 1/sqrt x using the power rule, we can use the following formula:

$\int x^ndx=\frac{x^{n+1}}{n+1}+c$

Since we need to calculate integral of 1/sqrt x, 

$\int \frac{1}{\sqrt{x}}dx=\int x^{-1/2} dx$

Now integrating by using the power rule of integral calculator,

$\int \frac{1}{\sqrt{x}}dx=\frac{x^{-1/2+1}{-1/2+1}+c$

$\int \frac{1}{\sqrt{x}}dx=\frac{x^{1/2}}{1/2}+c$

Or, 

$\int \frac{1}{\sqrt{x}}dx=2\sqrt{x}+c$

Hence we have verified the antiderivative of 1/sqrt x.

Integration of square root formula by using definite integral 

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)

Let’s understand the verification of the integral of ln by using the definite integral.

Proof of integral 1/root x by using definite integral

To integrate 1/sqrt x by using a definite integral calculator, we can use the interval from a to b. It means that we can evaluate the definite integral of 1/sqrtx for any value of a and b. Let’s compute it generally from a to b.

$∫^b_a \frac{1}{\sqrt{x}dx=2\sqrt{x}|^b_a$

Substituting the values of upper and lower bounds.

$∫^b_a \frac{1}{\sqrt{x}dx=2\sqrt{b}-2\sqrt{a}$

For any value of a and b, we can evaluate the definite integral by using the above formula. You can also use our definite integral calculator to make this calculation simple in just one click.

Frequently Asked Questions

How do you integrate square root of x?

To calculate the integral of the square root of x, we can use different integration techniques, such as integration by power rule, integration by parts and the u-substitution. Mathematically, the integral of sqrtx, is calculated as;

$\int \sqrt{x}dx={x^{1/2+1}}{1/2+1}=\frac{2x^{3/2}}{3}+c$

What is the integration of one by square root x?

The integration of one by square root x is equal to 2√x. It is expressed as;

$\int \frac{1}{\sqrt{x}}dx=\frac{x^{-1/2+1}}{-1/2+1}=2\sqrt{x}+c$